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Nahuatl has two sibilant fricatives, now pronounced something like [s] and [ʃ]. The standard orthography was developed by Spanish colonizers, who wrote /ʃ/ as x, and /s/ as c before a front vowel, z elsewhere. (There's also [t͡ʃ], written ch.)

But since all stages of Spanish definitely had s for [s], it seems clear that the sound the first transcribers heard definitely wasn't [s].

Do we know what this sound was? There unfortunately weren't trained linguists around transcribing Classical Nahuatl, but the Spanish transcription might be enough to make a good guess.

The reason that Spanish linguists transcribed the phoneme /s/ in Nahuatl as z/c(i/e) rather than s is because at that time Spanish had two alveolar sibilant phonemes, an apical /s̺/ written s and a laminal /s̻/ written z/c(i/e), and that the dental or alveolar sibilant in the Nahuatl spoken at that time was acoustically closer to the laminal than apical /s/. Since that time, in much of the Spanish spoken in Spain, /s̻/ has developed into /θ/, leaving only one alveolar sibilant.

There is a similar reason for why post-alveolar /ʃ/ is written x in the Classical Nahuatl orthography the Spanish introduced. At the time, in Early Modern Spanish, what is now for the most part written j and pronounced /x/ was actually pronounced /ʃ/ and written x. This later went on to merge with /ʒ/, written g(i/e).

all stages of Spanish definitely had s for [s], it seems clear that the sound the first transcribers heard definitely wasn't [s].

This seems like an oversimplification. Fricatives developed in different ways in different Spanish dialects: some had seseo (both <s> and <c/z> merged as [s]) some had ceceo (both <s> and <c/z> merged as [θ]) and some had distinction. The use of [θ] for <c/z> is thought to be a later development from some kind of sibilant distinct from the one used for <s>. The Iberian (but not Romance) language Basque still uses <z> for a voiceless sibilant with a quality distinct from <s>. IPA [s] isn't precise enough to notate the difference between the two Basque sounds: it's necessary to resort to diacritics. The IPA symbol [s] seems likely enough as a broad transcription of the Classical Nahuatl sound.